120 



Changes of a System of Revolution produced by Ordinary Refraction. 



10. Suppose now that the rays of this system of revolution fall 

 upon a refracting surface of revolution, having for axis the axis of 

 the system, and having for equation 



in which jj is still = .r* + ?/* = the square of the perpendicular distance 

 of a point xyz from the axis ; and let ;«.' be the refracting index of 

 the new medium into which the rays pass after refraction. It is evi- 

 dent that in this new medium, the rays will compose a new sysem of 

 revolution, symmetric about the same axis as before ; and we may in 

 general suppose the characteristic function V of this new system, 

 Avhich is analogous to V of the old, developed in a series similar 



to(P'), 



■*" 12(z + A'Y I z+ A' ■*" (2 -t- -dT 5 ■ 



the constants A' L' L\ being similar to A L L^ , in such a manner 

 that the ordinate Z' of intersection of the axis with a near ray, is 



Z' = — A'+L' («'^ + P) + L\ («'^ + fi'Y , (Y") 



if a* + /3'* denote the square of the sine of the angle which the near 

 ray makes with the axis, and if we neglect the sixth power of this 

 sine. To connect the new and old constants in the development of 

 the characteristic function, we have, by the nature of this function, 

 and by the principles of my former memoirs, the condition 



= AF=F'— F; (Z") 



